Yes, Alex, you can use a past participle after finire. An example from Devoto Oli:

Ha fatto tanto per gli altri per poi finire abbandonato

In this example andare wouldn't work, though I can't explain why -- maybe because its use is restricted to only a handful of verbs. I also get the impression that there is more of a passive meaning with andare (andarono distrutti = got destroyed), whereas with finire the emphasis is on the end result and the past participle is more like an adjective.